Blender 2.8: Camera Lens Shift and Background Images Offset
Blender: 2.81

I just installed the newly released blender 2.81 and my problem seems to be gone. Now, the background image always sticks centered to the camera (as expected), but you are still able to give a background image offset (x/y).

Blender: 2.8

However, I also figured out a solution for my problem in Blender 2.8. The offset depends on the camera resolution ratio. Assuming the render resolution X is larger than the render resolution y (e.g. 1920x1080), the math to solve my issue is:

For Background Image Offset X:


There is no magic
A background image offset X of 1 is a shift by the resolution width to the right. 
A background image offset X of -1 is a negative shift to the left.
The values between 0 and 1 are a fraction of the render resolution width, e.g. 0.1 = 10%, 0.5 = 50%, 0.75 = 75%, ...


For Background Image Offset Y:


This is a bit more complex, because the shift is related to the
render resolution
First, you need to divide the render resolution height by render resolution width to get a ratio (rr = rh / rw)
Second, you have to divide your camera lens shift y value by the calculated ratio (rr)
Third, you need to double this new value (for some reason?!)
Finally, you can set the newly calculated value to your background image offset y property



  Note: If the resolution height is larger than the resolution width, the solution from above is just flipped. Steps from Background Image OffsetX need to be done for Background Image OffsetY and vice versa.